Editing Exponential stability (section)

==Definition==
Consider the system
<math display="block">\dot{x} = f(t, x), \ x(t_0) = x_0,</math>
where <math display="inline">f</math> is piecewise continuous in <math display="inline">t</math> and [[Lipschitz continuity|Lipschitz]] in <math display="inline">x</math>. Assume without loss of generality that <math display="inline">f</math> has an [[Equilibrium point (mathematics)|equilibrium]] at the origin <math display="inline">x=0</math>. This equilibrium is exponentially stable if there exist <math display="inline">c, k, \lambda > 0</math> such that
<math display="block">
\| x(t) \| \leq k \| x(t_0) \| e^{-\lambda (t - t_0)},
</math>
for all <math display="inline> \| x(t_0) \| < c </math>.<ref>{{cite book |last1=Khalil |first1=Hassan |title=Nonlinear Systems |isbn=0-13-067389-7 |pages=150, 154}}</ref> That is, the solution <math display="inline">x(t)</math> displays an exponential rate of decay.